$\square \phi =f$

Let $V=\sqrt{|g|}$

$\frac{1}{V}(V{g}^{ij}{\phi }_{,i}{)}_{,j}=f$

...as finite difference...

$\frac{1}{V}{D}_j(V{g}^{ij}{D}_i\phi )=f$

expanding the inner finite difference term:

$\frac{1}{V[t,x,y,z]}($
$D_j(V[t,x,y,z]($
$\frac{1}{2\Delta t}{g}^{tj}[t,x,y,z](\phi [t+1,x,y,z]-\phi [t-1,x,y,z])$
$+\frac{1}{2\Delta x}{g}^{xj}[t,x,y,z](\phi [t,x+1,y,z]-\phi [t,x-1,y,z])$
$+\frac{1}{2\Delta y}{g}^{yj}[t,x,y,z](\phi [t,x,y+1,z]-\phi [t,x,y-1,z])$
$+\frac{1}{2\Delta z}{g}^{zj}[t,x,y,z](\phi [t,x,y,z+1]-\phi [t,x,y,z-1])$
$))$
$)=f$

expanding the outer finite difference terms:

$\frac{1}{V[t,x,y,z]}($
$\frac{1}{2\Delta t}($
$V[t+1,x,y,z]($
$\frac{1}{2\Delta t}{g}^{tt}[t+1,x,y,z](\phi [t+2,x,y,z]-\phi [t,x,y,z])$
$+\frac{1}{2\Delta x}{g}^{xt}[t+1,x,y,z](\phi [t+1,x+1,y,z]-\phi [t+1,x-1,y,z])$
$+\frac{1}{2\Delta y}{g}^{yt}[t+1,x,y,z](\phi [t+1,x,y+1,z]-\phi [t+1,x,y-1,z])$
$+\frac{1}{2\Delta z}{g}^{zt}[t+1,x,y,z](\phi [t+1,x,y,z+1]-\phi [t+1,x,y,z-1])$
$)-V[t-1,x,y,z]($
$\frac{1}{2\Delta t}{g}^{tt}[t-1,x,y,z](\phi [t,x,y,z]-\phi [t-2,x,y,z])$
$+\frac{1}{2\Delta x}{g}^{xt}[t-1,x,y,z](\phi [t-1,x+1,y,z]-\phi [t-1,x-1,y,z])$
$+\frac{1}{2\Delta y}{g}^{yt}[t-1,x,y,z](\phi [t-1,x,y+1,z]-\phi [t-1,x,y-1,z])$
$+\frac{1}{2\Delta z}{g}^{zt}[t-1,x,y,z](\phi [t-1,x,y,z+1]-\phi [t-1,x,y,z-1])$
$)$
$)$
$+\frac{1}{2\Delta x}($
$V[t,x+1,y,z]($
$\frac{1}{2\Delta t}{g}^{tx}[t,x+1,y,z](\phi [t+1,x+1,y,z]-\phi [t-1,x+1,y,z])$
$+\frac{1}{2\Delta x}{g}^{xx}[t,x+1,y,z](\phi [t,x+2,y,z]-\phi [t,x,y,z])$
$+\frac{1}{2\Delta y}{g}^{yx}[t,x+1,y,z](\phi [t,x+1,y+1,z]-\phi [t,x+1,y-1,z])$
$+\frac{1}{2\Delta z}{g}^{zx}[t,x+1,y,z](\phi [t,x+1,y,z+1]-\phi [t,x+1,y,z-1])$
$)-V[t,x-1,y,z]($
$\frac{1}{2\Delta t}{g}^{tx}[t,x-1,y,z](\phi [t+1,x-1,y,z]-\phi [t-1,x-1,y,z])$
$+\frac{1}{2\Delta x}{g}^{xx}[t,x-1,y,z](\phi [t,x,y,z]-\phi [t,x-2,y,z])$
$+\frac{1}{2\Delta y}{g}^{yx}[t,x-1,y,z](\phi [t,x-1,y+1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{2\Delta z}{g}^{zx}[t,x-1,y,z](\phi [t,x-1,y,z+1]-\phi [t,x-1,y,z-1])$
$)$
$)$
$+\frac{1}{2\Delta y}($
$V[t,x,y+1,z]($
$\frac{1}{2\Delta t}{g}^{ty}[t,x,y+1,z](\phi [t+1,x,y+1,z]-\phi [t-1,x,y+1,z])$
$+\frac{1}{2\Delta x}{g}^{xy}[t,x,y+1,z](\phi [t,x+1,y+1,z]-\phi [t,x-1,y+1,z])$
$+\frac{1}{2\Delta y}{g}^{yy}[t,x,y+1,z](\phi [t,x,y+2,z]-\phi [t,x,y,z])$
$+\frac{1}{2\Delta z}{g}^{zy}[t,x,y+1,z](\phi [t,x,y+1,z+1]-\phi [t,x,y+1,z-1])$
$)-V[t,x,y-1,z]($
$\frac{1}{2\Delta t}{g}^{ty}[t,x,y-1,z](\phi [t+1,x,y-1,z]-\phi [t-1,x,y-1,z])$
$+\frac{1}{2\Delta x}{g}^{xy}[t,x,y-1,z](\phi [t,x+1,y-1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{2\Delta y}{g}^{yy}[t,x,y-1,z](\phi [t,x,y,z]-\phi [t,x,y-2,z])$
$+\frac{1}{2\Delta z}{g}^{zy}[t,x,y-1,z](\phi [t,x,y-1,z+1]-\phi [t,x,y-1,z-1])$
$)$
$)$
$+\frac{1}{2\Delta z}($
$V[t,x,y,z+1]($
$\frac{1}{2\Delta t}{g}^{tz}[t,x,y,z+1](\phi [t+1,x,y,z+1]-\phi [t-1,x,y,z+1])$
$+\frac{1}{2\Delta x}{g}^{xz}[t,x,y,z+1](\phi [t,x+1,y,z+1]-\phi [t,x-1,y,z+1])$
$+\frac{1}{2\Delta y}{g}^{yz}[t,x,y,z+1](\phi [t,x,y+1,z+1]-\phi [t,x,y-1,z+1])$
$+\frac{1}{2\Delta z}{g}^{zz}[t,x,y,z+1](\phi [t,x,y,z+2]-\phi [t,x,y,z])$
$)-V[t,x,y,z-1]($
$\frac{1}{2\Delta t}{g}^{tz}[t,x,y,z-1](\phi [t+1,x,y,z-1]-\phi [t-1,x,y,z-1])$
$+\frac{1}{2\Delta x}{g}^{xz}[t,x,y,z-1](\phi [t,x+1,y,z-1]-\phi [t,x-1,y,z-1])$
$+\frac{1}{2\Delta y}{g}^{yz}[t,x,y,z-1](\phi [t,x,y+1,z-1]-\phi [t,x,y-1,z-1])$
$+\frac{1}{2\Delta z}{g}^{zz}[t,x,y,z-1](\phi [t,x,y,z]-\phi [t,x,y,z-2])$
$)$
$)$
$)=f$

solve for φ[t+2,x,y,z]

$\phi [t+2,x,y,z]$
$=\phi [t,x,y,z]$
$-\Delta t\frac{1}{g^{tt}[t+1,x,y,z]}($
$\frac{1}{\Delta x}{g}^{tx}[t+1,x,y,z](\phi [t+1,x+1,y,z]-\phi [t+1,x-1,y,z])$
$+\frac{1}{\Delta y}{g}^{ty}[t+1,x,y,z](\phi [t+1,x,y+1,z]-\phi [t+1,x,y-1,z])$
$+\frac{1}{\Delta z}{g}^{tz}[t+1,x,y,z](\phi [t+1,x,y,z+1]-\phi [t+1,x,y,z-1])$
$-\frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}($
$\frac{1}{\Delta t}{g}^{tt}[t-1,x,y,z](\phi [t,x,y,z]-\phi [t-2,x,y,z])$
$+\frac{1}{\Delta x}{g}^{tx}[t-1,x,y,z](\phi [t-1,x+1,y,z]-\phi [t-1,x-1,y,z])$
$+\frac{1}{\Delta y}{g}^{ty}[t-1,x,y,z](\phi [t-1,x,y+1,z]-\phi [t-1,x,y-1,z])$
$+\frac{1}{\Delta z}{g}^{tz}[t-1,x,y,z](\phi [t-1,x,y,z+1]-\phi [t-1,x,y,z-1])$
$)$
$)$
$-\frac{\Delta t^2}{V[t+1,x,y,z]g^{tt}[t+1,x,y,z]}($
$\frac{1}{\Delta x}($
$V[t,x+1,y,z]($
$\frac{1}{\Delta t}{g}^{tx}[t,x+1,y,z](\phi [t+1,x+1,y,z]-\phi [t-1,x+1,y,z])$
$+\frac{1}{\Delta x}{g}^{xx}[t,x+1,y,z](\phi [t,x+2,y,z]-\phi [t,x,y,z])$
$+\frac{1}{\Delta y}{g}^{xy}[t,x+1,y,z](\phi [t,x+1,y+1,z]-\phi [t,x+1,y-1,z])$
$+\frac{1}{\Delta z}{g}^{xz}[t,x+1,y,z](\phi [t,x+1,y,z+1]-\phi [t,x+1,y,z-1])$
$)-V[t,x-1,y,z]($
$\frac{1}{\Delta t}{g}^{tx}[t,x-1,y,z](\phi [t+1,x-1,y,z]-\phi [t-1,x-1,y,z])$
$+\frac{1}{\Delta x}{g}^{xx}[t,x-1,y,z](\phi [t,x,y,z]-\phi [t,x-2,y,z])$
$+\frac{1}{\Delta y}{g}^{xy}[t,x-1,y,z](\phi [t,x-1,y+1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{\Delta z}{g}^{xz}[t,x-1,y,z](\phi [t,x-1,y,z+1]-\phi [t,x-1,y,z-1])$
$)$
$)$
$+\frac{1}{\Delta y}($
$V[t,x,y+1,z]($
$\frac{1}{\Delta t}{g}^{ty}[t,x,y+1,z](\phi [t+1,x,y+1,z]-\phi [t-1,x,y+1,z])$
$+\frac{1}{\Delta x}{g}^{xy}[t,x,y+1,z](\phi [t,x+1,y+1,z]-\phi [t,x-1,y+1,z])$
$+\frac{1}{\Delta y}{g}^{yy}[t,x,y+1,z](\phi [t,x,y+2,z]-\phi [t,x,y,z])$
$+\frac{1}{\Delta z}{g}^{yz}[t,x,y+1,z](\phi [t,x,y+1,z+1]-\phi [t,x,y+1,z-1])$
$)-V[t,x,y-1,z]($
$\frac{1}{\Delta t}{g}^{ty}[t,x,y-1,z](\phi [t+1,x,y-1,z]-\phi [t-1,x,y-1,z])$
$+\frac{1}{\Delta x}{g}^{xy}[t,x,y-1,z](\phi [t,x+1,y-1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{\Delta y}{g}^{yy}[t,x,y-1,z](\phi [t,x,y,z]-\phi [t,x,y-2,z])$
$+\frac{1}{\Delta z}{g}^{yz}[t,x,y-1,z](\phi [t,x,y-1,z+1]-\phi [t,x,y-1,z-1])$
$)$
$)$
$+\frac{1}{\Delta z}($
$V[t,x,y,z+1]($
$\frac{1}{\Delta t}{g}^{tz}[t,x,y,z+1](\phi [t+1,x,y,z+1]-\phi [t-1,x,y,z+1])$
$+\frac{1}{\Delta x}{g}^{xz}[t,x,y,z+1](\phi [t,x+1,y,z+1]-\phi [t,x-1,y,z+1])$
$+\frac{1}{\Delta y}{g}^{yz}[t,x,y,z+1](\phi [t,x,y+1,z+1]-\phi [t,x,y-1,z+1])$
$+\frac{1}{\Delta z}{g}^{zz}[t,x,y,z+1](\phi [t,x,y,z+2]-\phi [t,x,y,z])$
$)-V[t,x,y,z-1]($
$\frac{1}{\Delta t}{g}^{tz}[t,x,y,z-1](\phi [t+1,x,y,z-1]-\phi [t-1,x,y,z-1])$
$+\frac{1}{\Delta x}{g}^{xz}[t,x,y,z-1](\phi [t,x+1,y,z-1]-\phi [t,x-1,y,z-1])$
$+\frac{1}{\Delta y}{g}^{yz}[t,x,y,z-1](\phi [t,x,y+1,z-1]-\phi [t,x,y-1,z-1])$
$+\frac{1}{\Delta z}{g}^{zz}[t,x,y,z-1](\phi [t,x,y,z]-\phi [t,x,y,z-2])$
$)$
$)$
$+\frac{1}{V[t,x,y,z]}f$
$)$

substitute 2012 Visser acoustic black hole metric:
$g^{tt}=-\frac{1}{c^2}$
$g^{ti}=-\frac{v^i}{c^2}$
$g^{ij}={\delta }^{ij}-\frac{v^i{v}^j}{c^2}$

$\phi [t+2,x,y,z]$
$=\phi [t,x,y,z]$
$+\Delta t($
$\frac{1}{\Delta t}\frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}(\phi [t,x,y,z]-\phi [t-2,x,y,z])$
$+\frac{1}{\Delta x}($
${\beta }^x[t+1,x,y,z](\phi [t+1,x+1,y,z]-\phi [t+1,x-1,y,z])$
$-{\beta }^x[t-1,x,y,z](\phi [t-1,x+1,y,z]-\phi [t-1,x-1,y,z])\frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$)$
$+\frac{1}{\Delta y}($
${\beta }^y[t+1,x,y,z](\phi [t+1,x,y+1,z]-\phi [t+1,x,y-1,z])$
$-{\beta }^y[t-1,x,y,z](\phi [t-1,x,y+1,z]-\phi [t-1,x,y-1,z])\frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$)$
$+\frac{1}{\Delta z}($
${\beta }^z[t+1,x,y,z](\phi [t+1,x,y,z+1]-\phi [t+1,x,y,z-1])$
$-{\beta }^z[t-1,x,y,z](\phi [t-1,x,y,z+1]-\phi [t-1,x,y,z-1])\frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$)$
$+\frac{\Delta t}{V[t+1,x,y,z]}(c[x,y,z]{)}^2($
$\frac{1}{\Delta x}($
$V[t,x+1,y,z]($
$\frac{1}{\Delta t}(\frac{{\beta }^x}{c^2})[t,x+1,y,z](\phi [t+1,x+1,y,z]-\phi [t-1,x+1,y,z])$
$+\frac{1}{\Delta x}{g}^{xx}[t,x+1,y,z](\phi [t,x+2,y,z]-\phi [t,x,y,z])$
$+\frac{1}{\Delta y}{g}^{xy}[t,x+1,y,z](\phi [t,x+1,y+1,z]-\phi [t,x+1,y-1,z])$
$+\frac{1}{\Delta z}{g}^{xz}[t,x+1,y,z](\phi [t,x+1,y,z+1]-\phi [t,x+1,y,z-1])$
$)-V[t,x-1,y,z]($
$\frac{1}{\Delta t}(\frac{{\beta }^x}{c^2})[t,x-1,y,z](\phi [t+1,x-1,y,z]-\phi [t-1,x-1,y,z])$
$+\frac{1}{\Delta x}{g}^{xx}[t,x-1,y,z](\phi [t,x,y,z]-\phi [t,x-2,y,z])$
$+\frac{1}{\Delta y}{g}^{xy}[t,x-1,y,z](\phi [t,x-1,y+1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{\Delta z}{g}^{xz}[t,x-1,y,z](\phi [t,x-1,y,z+1]-\phi [t,x-1,y,z-1])$
$)$
$)$
$+\frac{1}{\Delta y}($
$V[t,x,y+1,z]($
$\frac{1}{\Delta t}(\frac{{\beta }^y}{c^2})[t,x,y+1,z](\phi [t+1,x,y+1,z]-\phi [t-1,x,y+1,z])$
$+\frac{1}{\Delta x}{g}^{xy}[t,x,y+1,z](\phi [t,x+1,y+1,z]-\phi [t,x-1,y+1,z])$
$+\frac{1}{\Delta y}{g}^{yy}[t,x,y+1,z](\phi [t,x,y+2,z]-\phi [t,x,y,z])$
$+\frac{1}{\Delta z}{g}^{yz}[t,x,y+1,z](\phi [t,x,y+1,z+1]-\phi [t,x,y+1,z-1])$
$)-V[t,x,y-1,z]($
$\frac{1}{\Delta t}(\frac{{\beta }^y}{c^2})[t,x,y-1,z](\phi [t+1,x,y-1,z]-\phi [t-1,x,y-1,z])$
$+\frac{1}{\Delta x}{g}^{xy}[t,x,y-1,z](\phi [t,x+1,y-1,z]-\phi [t,x-1,y-1,z])$
$+\frac{1}{\Delta y}{g}^{yy}[t,x,y-1,z](\phi [t,x,y,z]-\phi [t,x,y-2,z])$
$+\frac{1}{\Delta z}{g}^{yz}[t,x,y-1,z](\phi [t,x,y-1,z+1]-\phi [t,x,y-1,z-1])$
$)$
$)$
$+\frac{1}{\Delta z}($
$V[t,x,y,z+1]($
$\frac{1}{\Delta t}(\frac{{\beta }^z}{c^2})[t,x,y,z+1](\phi [t+1,x,y,z+1]-\phi [t-1,x,y,z+1])$
$+\frac{1}{\Delta x}{g}^{xz}[t,x,y,z+1](\phi [t,x+1,y,z+1]-\phi [t,x-1,y,z+1])$
$+\frac{1}{\Delta y}{g}^{yz}[t,x,y,z+1](\phi [t,x,y+1,z+1]-\phi [t,x,y-1,z+1])$
$+\frac{1}{\Delta z}{g}^{zz}[t,x,y,z+1](\phi [t,x,y,z+2]-\phi [t,x,y,z])$
$)-V[t,x,y,z-1]($
$\frac{1}{\Delta t}(\frac{{\beta }^z}{c^2})[t,x,y,z-1](\phi [t+1,x,y,z-1]-\phi [t-1,x,y,z-1])$
$+\frac{1}{\Delta x}{g}^{xz}[t,x,y,z-1](\phi [t,x+1,y,z-1]-\phi [t,x-1,y,z-1])$
$+\frac{1}{\Delta y}{g}^{yz}[t,x,y,z-1](\phi [t,x,y+1,z-1]-\phi [t,x,y-1,z-1])$
$+\frac{1}{\Delta z}{g}^{zz}[t,x,y,z-1](\phi [t,x,y,z]-\phi [t,x,y,z-2])$
$)$
$)$
$+\frac{1}{V[t,x,y,z]}f$
$)$
$)$